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Properties

Label 2-384-12.11-c1-0-14
Degree 22
Conductor 384384
Sign −0.356+0.934i-0.356 + 0.934i
Analytic cond. 3.066253.06625
Root an. cond. 1.751071.75107
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.618 − 1.61i)3-s − 3.23i·5-s + 1.23i·7-s + (−2.23 − 2.00i)9-s + 5.23·11-s − 4.47·13-s + (−5.23 − 2.00i)15-s − 2.47i·17-s + 0.763i·19-s + (2.00 + 0.763i)21-s − 2.47·23-s − 5.47·25-s + (−4.61 + 2.38i)27-s − 4.76i·29-s + 5.23i·31-s + ⋯
L(s)  = 1  + (0.356 − 0.934i)3-s − 1.44i·5-s + 0.467i·7-s + (−0.745 − 0.666i)9-s + 1.57·11-s − 1.24·13-s + (−1.35 − 0.516i)15-s − 0.599i·17-s + 0.175i·19-s + (0.436 + 0.166i)21-s − 0.515·23-s − 1.09·25-s + (−0.888 + 0.458i)27-s − 0.884i·29-s + 0.940i·31-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=((−0.356+0.934i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+1/2) L(s)=((−0.356+0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: −0.356+0.934i-0.356 + 0.934i
Analytic conductor: 3.066253.06625
Root analytic conductor: 1.751071.75107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ384(383,⋅)\chi_{384} (383, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 384, ( :1/2), −0.356+0.934i)(2,\ 384,\ (\ :1/2),\ -0.356 + 0.934i)

Particular Values

L(1)L(1) ≈\approx 0.821977−1.19386i0.821977 - 1.19386i
L(12)L(\frac12) ≈\approx 0.821977−1.19386i0.821977 - 1.19386i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(−0.618+1.61i)T 1 + (-0.618 + 1.61i)T
good5 1+3.23iT−5T2 1 + 3.23iT - 5T^{2}
7 1−1.23iT−7T2 1 - 1.23iT - 7T^{2}
11 1−5.23T+11T2 1 - 5.23T + 11T^{2}
13 1+4.47T+13T2 1 + 4.47T + 13T^{2}
17 1+2.47iT−17T2 1 + 2.47iT - 17T^{2}
19 1−0.763iT−19T2 1 - 0.763iT - 19T^{2}
23 1+2.47T+23T2 1 + 2.47T + 23T^{2}
29 1+4.76iT−29T2 1 + 4.76iT - 29T^{2}
31 1−5.23iT−31T2 1 - 5.23iT - 31T^{2}
37 1−8.47T+37T2 1 - 8.47T + 37T^{2}
41 1−6.47iT−41T2 1 - 6.47iT - 41T^{2}
43 1+7.23iT−43T2 1 + 7.23iT - 43T^{2}
47 1−8T+47T2 1 - 8T + 47T^{2}
53 1+3.23iT−53T2 1 + 3.23iT - 53T^{2}
59 1+1.23T+59T2 1 + 1.23T + 59T^{2}
61 1−0.472T+61T2 1 - 0.472T + 61T^{2}
67 1−9.70iT−67T2 1 - 9.70iT - 67T^{2}
71 1−15.4T+71T2 1 - 15.4T + 71T^{2}
73 1+2T+73T2 1 + 2T + 73T^{2}
79 1−0.291iT−79T2 1 - 0.291iT - 79T^{2}
83 1−2.76T+83T2 1 - 2.76T + 83T^{2}
89 1−4iT−89T2 1 - 4iT - 89T^{2}
97 1−0.472T+97T2 1 - 0.472T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.59955937811697387787218550642, −9.700004049350334869849213063187, −9.123085472392491002397589723884, −8.375311004136550745503923058818, −7.38768480962131930711332998032, −6.31772620663705454161691159052, −5.24017022313894892514145531576, −4.06233545444411783821465339472, −2.34162863280202814876175064112, −0.990795964148147302576711378088, 2.41150850783916056386223581536, 3.59307890147807467427570262616, 4.39857266292059540737170685877, 5.95528125231371632951653869061, 6.93886039116066102818692117218, 7.81355643235460799238542660951, 9.169719782415862563373545761261, 9.835940482815470902539602454077, 10.66935905546697630113358430958, 11.31235256190131899215838706657

Graph of the ZZ-function along the critical line